(sometimes known as Riemannian geometry
) is a non-Euclidean geometry
, in which, given a line L
and a point p
, there exists no line parallel
passing through p
Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all. A simple way to picture this is to look at a globe. The lines of longitude are exactly next to each other, yet they eventually intersect. Elliptic geometry has other unusual properties. For example, the sum of the angles of any triangle is always greater than 180°.
Models of elliptic geometry
Models of elliptic geometry include the hyperspherical model, the projective model, and the stereographic model.
In the hyperspherical model, the points of n-dimensional elliptic space are the unit vectors in Rn+1, that is, the points on the surface of the unit ball in n+1 dimensional space. Lines in this model are great circles; intersections of the ball with hypersurface subspaces, meaning subspaces of dimension n.
In the projective model, the points of n-dimensional real projective space are used as points of the model. The points of n-dimensional projective space can be identified with lines through the origin in (n+1)-dimensional space, and can be represented non-uniquely by nonzero vectors in Rn+1, with the understanding that u and λu, for any non-zero scalar λ, represent the same point. Distance can be defined using the metric